Difference Methods for Double-Change Covering Designs

TL;DR

This paper advances the theory of double-change covering designs (DCCD) by developing two complementary construction paradigms: a recursive expansion-set approach that, together with 1-factorizations, builds larger DCCD from smaller ones while preserving minimality and circularity, and cyclic/difference-based constructions that yield tight circular DCCD across several parameter families. It establishes concrete existence results, including circular DCCD$(2k-2,k,k-1)$ for even $k$ and nonexistence for odd $k$, minimum circular DCCD$(2k-2,k,k)$ for odd $k\ge5$, and infinite families of tight circular DCCD$$(c(4k-6)+1,k,c^2(4k-6)+c)$$ for $1\le c\le5$, plus a notable example DCCD$(61,4,366)$. These contributions provide explicit, scalable methods to generate minimal or circular DCCD across a wide range of $v$, $k$, and $b$, enriching design theory and enabling practical applications in areas requiring efficient pair coverage with limited block changes. The work also highlights open directions, such as handling even expansion sizes in the recursive construction and extending constructions to larger $k$.

Abstract

A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366).

Figures (5)

• Figure 1: 1-factorization of $K_6$
• Figure 2: K$_8$
• Figure 3: Visualizing the cyclic difference construction for DCCD(11,4,11)
• Figure 4: $B_{0,0}$ and $B_{1,0}$
• Figure 5: $B_{1,0}$ and $B_{0,1}$

Theorems & Definitions (25)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Example 5
• Example 6